RÉSONATEURS NANOMÉCANIQUES DANS LE RÉGIME QUANTIQUE NANOMECHANICAL RESONATORS IN QUANTUM REGIME

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1 Chaire de Physique Mésoscopique Michel Devoret Année 1, 15 mai - 19 juin RÉSONATEURS NANOMÉCANIQUES DANS LE RÉGIME QUANTIQUE NANOMECHANICAL RESONATORS IN QUANTUM REGIME Troisième leçon / Third lecture This College de France document is for consultation only. Reproduction rights are reserved. Last update: III-1

2 PROGRAM OF THIS YEAR'S LECTURES Lecture I: Introduction to nanomechanical systems Lecture II: How do we model the coupling between electromagnetic modes and mechanical motion? Lecture III: Is zero-point motion of the detector variables a limitation in the measurement of the position of a mechanical oscillator? Lecture IV: In the active cooling of a nanomechanical resonator, is measurement-based feedback inferior to autonomous feedback? Lecture V: How close to the ground state can we bring a nanoresonator? Lecture VI: What oscillator characteristics must we choose to convert quantum information from the microwave domain to the optical domain? 1-III-

3 CALENDAR OF 1 SEMINARS May 15: Rob Schoelkopf (Yale University, USA) Quantum optics and quantum computation with superconducting circuits. May : Konrad Lehnert (JILA, Boulder, USA) Micro-electromechanics: a new quantum technology. May 9: Olivier Arcizet (Institut Néel, Grenoble) A single NV defect coupled to a nanomechanical oscillator: hybrid nanomechanics. June 5: Ivan Favero (MPQ, Université Paris Diderot) From micro to nano-optomechanical systems: photons interacting with mechanical resonators. June 1: A. Douglas Stone (Yale University, USA) Lasers and anti-lasers: a mesoscopic physicist s perspective on scattering from active and passive media. June 19: Tobias J. Kippenberg (EPFL, Lausanne, Suisse) Cavity optomechanics: exploring the coupling of light and micro- and nanomechanical oscillators. 1-III-3

4 LECTURE III : POSITION MEASUREMENT OF A MECHANICAL RESONATOR IN QUANTUM LIMIT OUTLINE 1. Quadrature representation of harmonic oscillator. Imprecision of single shot interference measurement of position 3. Continuous monitoring of position and associated backaction 1-III-4

5 Review: QUADRATURE REPRESENTATION OF A QUANTUM HARMONIC OSCILLATOR ω m ˆ ˆˆ 1 ˆˆ ωm ωm ˆ P K ˆ H = + X = b b+ ; = K ; b, b = 1 M M ˆ ( ˆ ˆ ) / X = XZPF b+ b ; XZPF = ; Zm = KM; PZPF = Z X Define dimens less hermitian operators*: m ˆ ˆ ˆ X, ˆ P I Q X P ZPF ZPF ZPF = + X ω e E zˆ X e Iˆ ˆ ˆ b+ b R ˆ, b Xφ = a.k.a. ( ) Qˆ ˆ ˆ b b i I b ˆ, ( ) a.k.a. Xφ = π / IQ ˆ, ˆ = i Δ = ˆ ˆ I I I Δ = ˆ ˆ Q Q Q ΔI ΔQ 1 4 standard deviations These definitions yield properties useful later. * see S. Haroche & J.M. Raimond, "Exploring the Quantum", Cambridge 6 1-III-5

6 COHERENT STATE IN QUADRATURE REPRESENTATION n β / β β = e n ; b β = β β ; β = e n! n= iωmt () t β ( ) ( ) ˆ ˆ ˆ i n= b b; β n β = n = β ; Δ n= β n n β = n; β = ne θ Q m Imβ "Fresnel lollypop" Δ I m = 1 Q m = Q m For coherent state: 1 Δ Im =Δ Qm = n θ Re β = I m I m Gaussian distribution of I m values ω m t I m 1-III-6

7 THE VACUUM STATE β = = n = Q m I m = Q = m ΔQ m Δ I =Δ Q = m m 1 I m ΔI m All coherent states can be thought of as a translated vacuum state in quadrature space. β = β + iβ Q m β β β I m Driving an harmonic oscillator with an arbitrary time-dependent force can only result in displacing the vacuum state. 1-III-7

8 FLYING OSCILLATOR A wave-packet propagating in medium with constant phase velocity can be seen as as oscillator Signal : (electric field, voltage, etc...) Ta A t t d Envelope ( ) varies slowly compared with center frequency () ( ) ω c Sˆ t = Env t t ˆ cos ˆ d S ZPF IS ωct+ QS sinωct k spatial mode ω temporal mode k c y ω c π T a t y d t d 1-III-8

9 HOMODYNE MEASUREMENT Measurement of a coherent state: α S Ideal photodetector ω c λ/4 α LO reference a.k.a. "local ω c 1-III-9

10 HOMODYNE MEASUREMENT Measurement of a coherent state: α S Ideal photodetector density matrix at input of beams: ˆ ρ = α, α α, α i LO S S LO α LO reference a.k.a. "local ω c aˆlo aˆ λ/4 aˆ = S iaˆlo nˆ Measurement of an arbitrary state: aˆs ( ˆ ) + i ia LO = aˆ aˆ aˆ 1 iaˆ = n LO =Tr + iaˆ S [ ˆ ρ nˆ ] i Output is D = α ( cos ˆ ˆ ) LO θl IS + sinθl QS ω c α LO αs αloα + S αloαs = nˆ = aˆ aˆ n =Tr [ ˆ ρ nˆ ] 1 i 1 α LO + αs + αloαs + αloαs = D ˆ = n ˆ n ˆ 1 D = α α + α α LO S LO S = α α + α α LO S LO S Ideal homodyne setup measures a generalized quadrature 1-III-9a

11 FLUCTUATIONS OF A HOMODYNE MEASUREMENT D ˆ = n ˆ n ˆ The photoelectron difference number is 1 We suppose that the efficiency of the detector is unity. ˆ 1 D= aˆ + aˆ aˆ + aˆ aˆ aˆ aˆ aˆ = + = α + δaˆ ( LO S )( LO S ) ( LO S )( LO S ) ( aˆ ˆ ˆ ˆ LOaS aloas ) aˆ ( α ˆ ˆ ˆ ˆ ˆ ˆ LOaS αloas δaloas δaloas ) = Fluctuations are of order N 1/ Fluctuations are of order unity We have thus, in the limit of large LO number of photons: ( ˆ θ ˆ ) θ Dˆ N I cos + Q sin LO S L S L LO LO LO 1-III-1

12 FLUCTUATIONS OF A HOMODYNE MEASUREMENT D ˆ = n ˆ n ˆ The photoelectron difference number is 1 We suppose that the efficiency of the detector is unity. ˆ 1 D= aˆ + aˆ aˆ + aˆ aˆ aˆ aˆ aˆ ( LO S )( LO S ) ( LO S )( LO S ) ( aˆ ˆ ˆ ˆ LOaS aloas ) ( α ˆ ˆ ˆ ˆ ˆ ˆ LOaS αloas δaloas δaloas ) = + = aˆ = α + δaˆ LO LO LO Fluctuations are of order N 1/ Fluctuations are of order unity We have thus, in the limit of large LO number of photons: Probability P( x S ) For a coherent state, =1 ( ˆ θ ˆ ) θ Dˆ N I cos + Q sin LO S L S L ΔxS cosθ Iˆ + sinθ Qˆ L S L S detector inefficiency finite LO strength result in a further broadening of this distribution. x S = D N LO 1-III-1a

13 HOMODYNE MEASUREMENT PERFORMS A PROJECTION IN QUADRATURE PLANE Q S signal state x S = D N LO cosθ Iˆ + sinθ Qˆ L S L S θ L I S In the ideal case, homodyne measurement performs a noise-less measurement of an oscillator generalized quadrature. ( ) ( ˆ ˆ ) S = Tr ρ cosθl S + sinθl S P x I Q x ˆS ( θ ) LO 1-III-11

14 INTERFEROMETRIC MEASUREMENT OF POSITION = + X ( t) Single shot k e T a ω c 1 ω m e = 1 nx ZPF α = φ Ne iθ αe iδθ ω m simplified representation of separation of input and output beams Q S δ I S = Nδθ δθ = keδx = 4kX δ I I S e ZPF m 1 4 NkeXZPF δ I m () ˆ () Dˆ = α ˆ cos ˆ sin ˆ LO IS ke X t + QS ke X t α ˆ 4 ˆ ˆ LO IS + ke XZPFQS Im () t Convolution of two fluctuations: mechanical and probe photon shot noise. x = D/8 N Nk X m LO e ZPF 1-III-1

15 IS IT POSSIBLE TO MAKE PHOTON SHOT NOISE NEGLIGIBLE COMPARED TO POSITION FLUCTUATIONS? IN OTHER WORDS, CAN WE HAVE 4 1 Nk X e ZPF δ I m X ZPF OR EVEN N 1 λ e 15 DIFFICULT SINCE 1 m X ZPF π X ZPF LAMB-DICKE PARAMETER kx e ZPF = ALWAYS SMALL! λ e 1-III-13

16 TWO HELPFUL FACTORS: - CAVITY (ELECTROMAGNETIC RESONATOR) - MANY PHOTONS (LONG ACQUISITION TIME) 1-III-14

17 ENHANCEMENT OF INTERFEROMETRIC PHASE-SHIFT BY CAVITY κ ( ) = + X t 1-III-15 κ 1 1 T a ω m α iθ αe r ω e ω m 3 ( ω) θr π κ D 1 c L cav ω 1 κ F = ω 1 κ see Cohadon,. Heidmann, and M. Pinard, PRL83, 3177(1999) ω e ω ˆ ˆ δ X δθ X = 8F λe X = 8F λ ZPF e ( b ˆ + b ) ˆ Enhancement of Lamb-Dicke parameter by finesse of cavity

18 μwave vs OPTICS LASER κ ω e ω m output κ μwave generator(s) φ ω e ω m L.O. output L.O. 1-III-16

19 MIXERS AND PHASE-SENSITIVE AMPLIFIERS signal L.O. DC DC Mixers have conversion gains less than unity. Add at least 3dB of noise. Following amplifier adds also noise. signal φ L.O. L.O. DC Josephson parametric amplifiers in phase sensitive mode amplify without adding noise 1 quadrature of signal. Has enough gain to beat noise of following amplifier. 1-III-17

20 SPECTRAL DENSITY OF OSCILLATOR MOTION Review: XX 1 X t X e d π 1 + iωt X = X t e dt π + iωt + () = [ ω] ω [ ω] () + [ ] ˆ () ˆ + iωt ω = ( ) ω ˆ [ ω] ˆ [ ω ] = [ ω] δ ( ω+ ω ) S X t X e d "Engineer" spectral density X temporal and ensemble averaging ~ γ 1 ~ n S X X S XX ω f = = S + S π [ ω] [ ω] XX XX XX t measured by usual spectrum analyzer X ( t ) = t ω m S XX [ ω] ~ n ~ ( n + 1) γ +ω m Total area under curve ~ number of phonons in mechanical resonator ω 1-III-18

21 SPECTRAL DENSITY OF HOMODYNE SIGNAL We now consider a continuous homodyne measurement. Q S δθ I S () t = FX ( t) 8 / λe NLO N N N D θ Dt θ xm = XZPF xm () t = XZPF 4 N X LON 4 N X LO N ( ) LO 1 1 ( ) S xx ω Apparent oscillator motion in quadrature representation γ 1 n + γ ω m ω Spectral density of oscillator apparent motion if photon shot noise is negligible, as well as backaction. 1-III-19

22 MEASUR ED NOISE: IMPRECISION AND BACKACTION ( ) tot S xx ω backaction S tot xx ( ω ) m (log scale) SQL total measured noise imprecision intrinsic noise imprecision backaction ω m ω zero-point fluctuations of resonator number of photons in cavity (log scale) N Standard Quantum Limit (SQL): optimal compromise between imprecision and backaction noises At SQL, the total noise energy in the resonator is equivalent to a full phonon. 1-III-1

23 IMPRECISION vs RESOLUTION 1) Measure fairness of coin by tossing it many times 5 trials will determine fairness with 1% imprecision (@ 1 standard deviation) ) Measure wavelength of incoming beam width of slit determines resolution 1-III-

24 END OF LECTURE